Inversion of the Multiplicative Matrix Compound Operator

We study the problem of determining a matrix whose $k$th multiplicative compound, with $k>1$, is a prescribed matrix $M$. The cardinality of the set of matrices whose $k$th multiplicative compound equals $M$ is characterized in terms of $\text{rank}(M)$. On the one hand, if $\text{rank}(M)\le 1$, it is shown that there exist infinitely many such matrices for which a complete characterization is determined. On the other hand, if $\text{rank}(M)>1$, then there exists a unique matrix – up to an overall sign – whose compound is $M$. An algorithm for finding a matrix whose compound equals $M$ is detailed, and its time complexity is analyzed.